Spatial structure and diffusive dynamics from single-particle trajectories using spline analysis

Abstract

Single-particle tracking of biomolecular probes has provided a wealth of information about intracellular trafficking and the dynamics of proteins and lipids in the cell membrane. Conventional mean-square displacement (MSD) analysis of single-particle trajectories often assumes that probes are moving in a uniform environment. However, the observed two-dimensional motion of probe particles is influenced by the local three-dimensional geometry of the cell membrane and intracellular structures, which are rarely flat at the submicron scale. This complex geometry can lead to spatially confined trajectories that are difficult to analyze and interpret using conventional two-dimensional MSD analysis. Here we present two methods to analyze spatially confined trajectories: spline-curve dynamics analysis, which extends conventional MSD analysis to measure diffusive motion in confined trajectories; and spline-curve spatial analysis, which measures spatial structures smaller than the limits of optical resolution. We show, using simulated random walks and experimental trajectories of quantum dot probes, that differences in measured two-dimensional diffusion coefficients do not always reflect differences in underlying diffusive dynamics, but can instead be due to differences in confinement geometries of cellular structures.

Figure 1

Trajectories A–D plotted with spline curves. (Top left) Trajectory A is a random walk simulated to model experimental trajectory B. (Top right) Trajectory B is an experimental trajectory showing two-dimensional diffusion that is easily analyzed with conventional two-dimensional MSD analysis. (Bottom left) Trajectory C shows linear, confined motion that is more difficult to analyze using conventional two-dimensional MSD. (Bottom right) Trajectory D shows spatial confinement to a curvilinear structure, which is ill suited for conventional two-dimensional MSD analysis, but ideal for the spline-curve analysis techniques discussed in the text. Trajectories B–D were collected from movement of QD probes incubated with PC12 cells (see Methods).

Figure 2

(Top) Spline-curve dynamics analysis (SCDA). A spline curve (dashed line) is fitted through the overall trajectory shape and each displacement Δ is resolved into a component parallel to the spline curve Δ_‖ and a component perpendicular to the spline curve Δ_⊥. The parallel and perpendicular displacements can then be analyzed independently to measure dynamics parallel and perpendicular to the spline curve. (Bottom) Spline-curve spatial analysis (SCSA). The distance d_i from each particle to the spline curve is calculated and histograms of d_i provide a spatial profile of the trajectory.

Figure 3

Comparison of conventional two-dimensional MSD analysis and SCDA. (Top left) Conventional two-dimensional MSD plotted for trajectories A–D. Conventional two-dimensional MSD characterizes trajectories C and D with smaller slope (lower D_2D) than that of experimental trajectory B and simulated random walk A, due to the spatial confinement in trajectories C and D. Conventional two-dimensional MSD is plotted here as (1/2) MSD to facilitate comparison with MSD_‖ and MSD_⊥. (Top right) SCDA applied to trajectories A and B shows similar MSD_‖ and MSD_⊥, confirming that the diffusive dynamics of A and B are isotropic and independent of the spline curve. (Bottom left) Trajectories C and D exhibit linear MSD_‖ with nearly identical slopes, showing similar free diffusion along the spline curve. However, MSD_⊥ is severely restricted because the trajectories are spatially confined to be near the spline curve. (Bottom right) Linear MSD_‖ with similar slopes for A–D, showing that all four trajectories have similar diffusive dynamics parallel to their spline curves (D_‖ values are given in Table 1). Error bars in all plots represent the statistical error due to finite sampling (Eq. 3).

Figure 4

Distributions of diffusion coefficients D_2D, D_‖, and D_⊥ for free diffusion, confined diffusion, and curvilinear SPT trajectories. (Left) D_2D, D_‖, and D_⊥ from 500 two-dimensional random walk trajectories simulating purely diffusive motion. D_‖ and D_⊥ reproduce the diffusion coefficient D_2D obtained using conventional two-dimensional MSD analysis. (Center) D_2D, D_‖, and D_⊥ from the two-dimensional projections of 500 two-dimensional random walk trajectories confined to a 100-nm radius cylinder. D_2D systematically underestimates the true diffusion coefficient, D = 0.24 μm²/s. D_‖ provides a more accurate measure of D because the unconfined motion along the spline curve is separated from the confined motion perpendicular to the spline curve. (Right) D_2D, D_‖, and D_⊥ from 159 curvilinear SPT trajectories. The range of D_‖ values includes a wide range of larger values than those reported by D_2D, illustrating that conventional two-dimensional MSD also underestimates diffusion in experimental trajectories.

Figure 5

Results of SCSA showing the distribution of distances from the spline curve for trajectories A–D as indicated. Width indicated is the standard deviation about the mean and N is the number of data points. Trajectories A and B show broad spatial distributions (top), whereas C and D show clear confinement near the fitted spline curve (bottom). Additionally, trajectory D exhibits a bimodal distribution, consistent with trajectory positions distributed on a cylindrical surface projected onto the image plane, as described in Results and Fig. 5.

Figure 6

(Top) Simple model of the distribution of measured position locations on a cylindrical shell. The cylinder has radius r = 90 nm and thickness 3 nm, with the bottom 3 nm excluded to model the region in contact with a surface. The density of measured positions is normal-distributed in the x direction corresponding to position measurement error of σ_m ± 27 nm. (Bottom) Projection of this cylindrical distribution onto the x axis, displaying a bimodal distribution similar to that seen in the spatial distribution of trajectory D (Fig. 5, bottom right).